Use this theorem to learn if there can be no solution:

Given positive number N, if positive numbers a and b exist such that
ab = N then the minimum value of a+b is 

We calculate  which is the smallest
value any sum the two positive numbers can have which have product 1440.

But 13 is less than that, so there is no possible solution.

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Then it occurred to me that maybe you meant "subtracted" instead of
"added together", for there is a solution to this question:

What two positive integers multiplied together gives me 1440, and subtracted
(larger - smaller) gives me 13?

To find out the minimum value of the minimum difference:

 1. First find the square root of 1440. We get 37.94733192
 2. Think of the closest factor of 1440 less than that, which is 36
 3. Divide 1440 by 36, getting 40
 4. Subtract them, getting 4.
 5. That's the minimum possible difference.
 6. Since 13 is greater than 4, there may be a solution.
 7. Get the next smaller factor of 1440 than 36, which is 32.
 8. Divide 1440 by 32, getting 45
10. Subtract them, getting 13.

That's it.  45×32 = 1440, and 45-32 = 13

So if you meant "subtracted (larger minus smaller)" instead of "added",
this is the answer.  But if you meant "added" there is no solution.

Edwin